Geometry

Triangle Centers

Triangle Centers: Level 2 Challenges

         

At the 1920 FIFA World Cup, there was an earth quake during one of the games and a cone shaped hole was created with a diameter of 4 feet and sloping sides, both 4 feet long from ground-level to the bottom of the pit. The soccer ball, with a radius of one foot, fell into the hole. Find the distance from the center of the Soccer ball to the bottom of the pit in feet.

Assumptions:
- The sides of the hole are perfectly straight and smooth.
- The ball falls as far as it can with out changing shape.
- FYI, there wasn't actually an earthquake at the 1920 FIFA World Cup.

In triangle ABCABC with centroid GG, if AG=BC AG=BC, what is angle BGCBGC in degrees?

Note: Diagram is not drawn to scale.

A triangle has sides of 6, 8, and 10 inches.

What is the distance between incenter and circumcenter of the triangle?

In the diagram above, line ll passes through the centroid of ABC.\triangle ABC.

If the perpendicular distance between AA and line ll is 2, and the perpendicular distance between BB and line ll is 6, then what is the perpendicular distance between CC and line l?l?

Consider an isosceles ABC\triangle ABC with AB=AC=5,BC=6,AB=AC=5, BC=6, where I,O,HI,O,H denote its incenter, circumcenter, orthocenter, respectively.

Find the area of IOH\triangle IOH.

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